Theorem pnt 24315

Description: The Prime Number Theorem: the number of prime numbers less than x tends asymptotically to x / log ( x ) as x goes to infinity. This is Metamath 100 proof #5. (Contributed by Mario Carneiro, 1-Jun-2016.)

Assertion

Ref	Expression
pnt	|- ( x e. ( 1 (,) +oo ) |-> ( (π ` x ) / ( x / ( log ` x ) ) ) ) ~~> r 1

Proof of Theorem pnt

Dummy variables w y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		1re 9641	. . . . . . 7 |- 1 e. RR
2	1	rexri 9692	. . . . . 6 |- 1 e. RR*
3		1lt2 10776	. . . . . 6 |- 1 < 2
4		df-ioo 11639	. . . . . . 7 |- (,) = ( x e. RR* , y e. RR* |-> { z e. RR* | ( x < z /\ z < y ) } )
5		df-ico 11641	. . . . . . 7 |- [,) = ( x e. RR* , y e. RR* |-> { z e. RR* | ( x <_ z /\ z < y ) } )
6		xrltletr 11454	. . . . . . 7 |- ( ( 1 e. RR* /\ 2 e. RR* /\ w e. RR* ) -> ( ( 1 < 2 /\ 2 <_ w ) -> 1 < w ) )
7	4, 5, 6	ixxss1 11653	. . . . . 6 |- ( ( 1 e. RR* /\ 1 < 2 ) -> ( 2 [,) +oo ) C_ ( 1 (,) +oo ) )
8	2, 3, 7	mp2an 676	. . . . 5 |- ( 2 [,) +oo ) C_ ( 1 (,) +oo )
9		resmpt 5174	. . . . 5 |- ( ( 2 [,) +oo ) C_ ( 1 (,) +oo ) -> ( ( x e. ( 1 (,) +oo ) |-> ( (π ` x ) / ( x / ( log ` x ) ) ) ) |` ( 2 [,) +oo ) ) = ( x e. ( 2 [,) +oo ) |-> ( (π ` x ) / ( x / ( log ` x ) ) ) ) )
10	8, 9	mp1i 13	. . . 4 |- ( T. -> ( ( x e. ( 1 (,) +oo ) |-> ( (π ` x ) / ( x / ( log ` x ) ) ) ) |` ( 2 [,) +oo ) ) = ( x e. ( 2 [,) +oo ) |-> ( (π ` x ) / ( x / ( log ` x ) ) ) ) )
11	8	sseli 3466	. . . . . . . . . 10 |- ( x e. ( 2 [,) +oo ) -> x e. ( 1 (,) +oo ) )
12		ioossre 11696	. . . . . . . . . . 11 |- ( 1 (,) +oo ) C_ RR
13	12	sseli 3466	. . . . . . . . . 10 |- ( x e. ( 1 (,) +oo ) -> x e. RR )
14	11, 13	syl 17	. . . . . . . . 9 |- ( x e. ( 2 [,) +oo ) -> x e. RR )
15		2re 10679	. . . . . . . . . . 11 |- 2 e. RR
16		pnfxr 11412	. . . . . . . . . . 11 |- +oo e. RR*
17		elico2 11698	. . . . . . . . . . 11 |- ( ( 2 e. RR /\ +oo e. RR* ) -> ( x e. ( 2 [,) +oo ) <-> ( x e. RR /\ 2 <_ x /\ x < +oo ) ) )
18	15, 16, 17	mp2an 676	. . . . . . . . . 10 |- ( x e. ( 2 [,) +oo ) <-> ( x e. RR /\ 2 <_ x /\ x < +oo ) )
19	18	simp2bi 1021	. . . . . . . . 9 |- ( x e. ( 2 [,) +oo ) -> 2 <_ x )
20		chtrpcl 23965	. . . . . . . . 9 |- ( ( x e. RR /\ 2 <_ x ) -> ( theta ` x ) e. RR+ )
21	14, 19, 20	syl2anc 665	. . . . . . . 8 |- ( x e. ( 2 [,) +oo ) -> ( theta ` x ) e. RR+ )
22		0red 9643	. . . . . . . . . . 11 |- ( x e. ( 1 (,) +oo ) -> 0 e. RR )
23	1	a1i 11	. . . . . . . . . . 11 |- ( x e. ( 1 (,) +oo ) -> 1 e. RR )
24		0lt1 10135	. . . . . . . . . . . 12 |- 0 < 1
25	24	a1i 11	. . . . . . . . . . 11 |- ( x e. ( 1 (,) +oo ) -> 0 < 1 )
26		eliooord 11694	. . . . . . . . . . . 12 |- ( x e. ( 1 (,) +oo ) -> ( 1 < x /\ x < +oo ) )
27	26	simpld 460	. . . . . . . . . . 11 |- ( x e. ( 1 (,) +oo ) -> 1 < x )
28	22, 23, 13, 25, 27	lttrd 9795	. . . . . . . . . 10 |- ( x e. ( 1 (,) +oo ) -> 0 < x )
29	13, 28	elrpd 11338	. . . . . . . . 9 |- ( x e. ( 1 (,) +oo ) -> x e. RR+ )
30	11, 29	syl 17	. . . . . . . 8 |- ( x e. ( 2 [,) +oo ) -> x e. RR+ )
31	21, 30	rpdivcld 11358	. . . . . . 7 |- ( x e. ( 2 [,) +oo ) -> ( ( theta ` x ) / x ) e. RR+ )
32	31	adantl 467	. . . . . 6 |- ( ( T. /\ x e. ( 2 [,) +oo ) ) -> ( ( theta ` x ) / x ) e. RR+ )
33		ppinncl 23964	. . . . . . . . . . 11 |- ( ( x e. RR /\ 2 <_ x ) -> (π ` x ) e. NN )
34	14, 19, 33	syl2anc 665	. . . . . . . . . 10 |- ( x e. ( 2 [,) +oo ) -> (π ` x ) e. NN )
35	34	nnrpd 11339	. . . . . . . . 9 |- ( x e. ( 2 [,) +oo ) -> (π ` x ) e. RR+ )
36	13, 27	rplogcld 23443	. . . . . . . . . 10 |- ( x e. ( 1 (,) +oo ) -> ( log ` x ) e. RR+ )
37	11, 36	syl 17	. . . . . . . . 9 |- ( x e. ( 2 [,) +oo ) -> ( log ` x ) e. RR+ )
38	35, 37	rpmulcld 11357	. . . . . . . 8 |- ( x e. ( 2 [,) +oo ) -> ( (π ` x ) x. ( log ` x ) ) e. RR+ )
39	21, 38	rpdivcld 11358	. . . . . . 7 |- ( x e. ( 2 [,) +oo ) -> ( ( theta ` x ) / ( (π ` x ) x. ( log ` x ) ) ) e. RR+ )
40	39	adantl 467	. . . . . 6 |- ( ( T. /\ x e. ( 2 [,) +oo ) ) -> ( ( theta ` x ) / ( (π ` x ) x. ( log ` x ) ) ) e. RR+ )
41	30	ssriv 3474	. . . . . . . 8 |- ( 2 [,) +oo ) C_ RR+
42		resmpt 5174	. . . . . . . 8 |- ( ( 2 [,) +oo ) C_ RR+ -> ( ( x e. RR+ |-> ( ( theta ` x ) / x ) ) |` ( 2 [,) +oo ) ) = ( x e. ( 2 [,) +oo ) |-> ( ( theta ` x ) / x ) ) )
43	41, 42	ax-mp 5	. . . . . . 7 |- ( ( x e. RR+ |-> ( ( theta ` x ) / x ) ) |` ( 2 [,) +oo ) ) = ( x e. ( 2 [,) +oo ) |-> ( ( theta ` x ) / x ) )
44		pnt2 24314	. . . . . . . 8 |- ( x e. RR+ |-> ( ( theta ` x ) / x ) ) ~~> r 1
45		rlimres 13600	. . . . . . . 8 |- ( ( x e. RR+ |-> ( ( theta ` x ) / x ) ) ~~> r 1 -> ( ( x e. RR+ |-> ( ( theta ` x ) / x ) ) |` ( 2 [,) +oo ) ) ~~> r 1 )
46	44, 45	mp1i 13	. . . . . . 7 |- ( T. -> ( ( x e. RR+ |-> ( ( theta ` x ) / x ) ) |` ( 2 [,) +oo ) ) ~~> r 1 )
47	43, 46	syl5eqbrr 4460	. . . . . 6 |- ( T. -> ( x e. ( 2 [,) +oo ) |-> ( ( theta ` x ) / x ) ) ~~> r 1 )
48		chtppilim 24176	. . . . . . 7 |- ( x e. ( 2 [,) +oo ) |-> ( ( theta ` x ) / ( (π ` x ) x. ( log ` x ) ) ) ) ~~> r 1
49	48	a1i 11	. . . . . 6 |- ( T. -> ( x e. ( 2 [,) +oo ) |-> ( ( theta ` x ) / ( (π ` x ) x. ( log ` x ) ) ) ) ~~> r 1 )
50		ax-1ne0 9607	. . . . . . 7 |- 1 =/= 0
51	50	a1i 11	. . . . . 6 |- ( T. -> 1 =/= 0 )
52	39	rpne0d 11346	. . . . . . 7 |- ( x e. ( 2 [,) +oo ) -> ( ( theta ` x ) / ( (π ` x ) x. ( log ` x ) ) ) =/= 0 )
53	52	adantl 467	. . . . . 6 |- ( ( T. /\ x e. ( 2 [,) +oo ) ) -> ( ( theta ` x ) / ( (π ` x ) x. ( log ` x ) ) ) =/= 0 )
54	32, 40, 47, 49, 51, 53	rlimdiv 13687	. . . . 5 |- ( T. -> ( x e. ( 2 [,) +oo ) |-> ( ( ( theta ` x ) / x ) / ( ( theta ` x ) / ( (π ` x ) x. ( log ` x ) ) ) ) ) ~~> r ( 1 / 1 ) )
55	14	recnd 9668	. . . . . . . . . 10 |- ( x e. ( 2 [,) +oo ) -> x e. CC )
56		chtcl 23899	. . . . . . . . . . . . 13 |- ( x e. RR -> ( theta ` x ) e. RR )
57	13, 56	syl 17	. . . . . . . . . . . 12 |- ( x e. ( 1 (,) +oo ) -> ( theta ` x ) e. RR )
58	57	recnd 9668	. . . . . . . . . . 11 |- ( x e. ( 1 (,) +oo ) -> ( theta ` x ) e. CC )
59	11, 58	syl 17	. . . . . . . . . 10 |- ( x e. ( 2 [,) +oo ) -> ( theta ` x ) e. CC )
60	55, 59	mulcomd 9663	. . . . . . . . 9 |- ( x e. ( 2 [,) +oo ) -> ( x x. ( theta ` x ) ) = ( ( theta ` x ) x. x ) )
61	60	oveq2d 6321	. . . . . . . 8 |- ( x e. ( 2 [,) +oo ) -> ( ( ( theta ` x ) x. ( (π ` x ) x. ( log ` x ) ) ) / ( x x. ( theta ` x ) ) ) = ( ( ( theta ` x ) x. ( (π ` x ) x. ( log ` x ) ) ) / ( ( theta ` x ) x. x ) ) )
62	38	rpcnd 11343	. . . . . . . . 9 |- ( x e. ( 2 [,) +oo ) -> ( (π ` x ) x. ( log ` x ) ) e. CC )
63	30	rpne0d 11346	. . . . . . . . 9 |- ( x e. ( 2 [,) +oo ) -> x =/= 0 )
64	21	rpne0d 11346	. . . . . . . . 9 |- ( x e. ( 2 [,) +oo ) -> ( theta ` x ) =/= 0 )
65	62, 55, 59, 63, 64	divcan5d 10408	. . . . . . . 8 |- ( x e. ( 2 [,) +oo ) -> ( ( ( theta ` x ) x. ( (π ` x ) x. ( log ` x ) ) ) / ( ( theta ` x ) x. x ) ) = ( ( (π ` x ) x. ( log ` x ) ) / x ) )
66	61, 65	eqtrd 2470	. . . . . . 7 |- ( x e. ( 2 [,) +oo ) -> ( ( ( theta ` x ) x. ( (π ` x ) x. ( log ` x ) ) ) / ( x x. ( theta ` x ) ) ) = ( ( (π ` x ) x. ( log ` x ) ) / x ) )
67	38	rpne0d 11346	. . . . . . . 8 |- ( x e. ( 2 [,) +oo ) -> ( (π ` x ) x. ( log ` x ) ) =/= 0 )
68	59, 55, 59, 62, 63, 67, 64	divdivdivd 10429	. . . . . . 7 |- ( x e. ( 2 [,) +oo ) -> ( ( ( theta ` x ) / x ) / ( ( theta ` x ) / ( (π ` x ) x. ( log ` x ) ) ) ) = ( ( ( theta ` x ) x. ( (π ` x ) x. ( log ` x ) ) ) / ( x x. ( theta ` x ) ) ) )
69	34	nncnd 10625	. . . . . . . 8 |- ( x e. ( 2 [,) +oo ) -> (π ` x ) e. CC )
70	37	rpcnd 11343	. . . . . . . 8 |- ( x e. ( 2 [,) +oo ) -> ( log ` x ) e. CC )
71	37	rpne0d 11346	. . . . . . . 8 |- ( x e. ( 2 [,) +oo ) -> ( log ` x ) =/= 0 )
72	69, 55, 70, 63, 71	divdiv2d 10414	. . . . . . 7 |- ( x e. ( 2 [,) +oo ) -> ( (π ` x ) / ( x / ( log ` x ) ) ) = ( ( (π ` x ) x. ( log ` x ) ) / x ) )
73	66, 68, 72	3eqtr4d 2480	. . . . . 6 |- ( x e. ( 2 [,) +oo ) -> ( ( ( theta ` x ) / x ) / ( ( theta ` x ) / ( (π ` x ) x. ( log ` x ) ) ) ) = ( (π ` x ) / ( x / ( log ` x ) ) ) )
74	73	mpteq2ia 4508	. . . . 5 |- ( x e. ( 2 [,) +oo ) |-> ( ( ( theta ` x ) / x ) / ( ( theta ` x ) / ( (π ` x ) x. ( log ` x ) ) ) ) ) = ( x e. ( 2 [,) +oo ) |-> ( (π ` x ) / ( x / ( log ` x ) ) ) )
75		1div1e1 10299	. . . . 5 |- ( 1 / 1 ) = 1
76	54, 74, 75	3brtr3g 4457	. . . 4 |- ( T. -> ( x e. ( 2 [,) +oo ) |-> ( (π ` x ) / ( x / ( log ` x ) ) ) ) ~~> r 1 )
77	10, 76	eqbrtrd 4446	. . 3 |- ( T. -> ( ( x e. ( 1 (,) +oo ) |-> ( (π ` x ) / ( x / ( log ` x ) ) ) ) |` ( 2 [,) +oo ) ) ~~> r 1 )
78		ppicl 23921	. . . . . . . . . 10 |- ( x e. RR -> (π ` x ) e. NN0 )
79	13, 78	syl 17	. . . . . . . . 9 |- ( x e. ( 1 (,) +oo ) -> (π ` x ) e. NN0 )
80	79	nn0red 10926	. . . . . . . 8 |- ( x e. ( 1 (,) +oo ) -> (π ` x ) e. RR )
81	29, 36	rpdivcld 11358	. . . . . . . 8 |- ( x e. ( 1 (,) +oo ) -> ( x / ( log ` x ) ) e. RR+ )
82	80, 81	rerpdivcld 11369	. . . . . . 7 |- ( x e. ( 1 (,) +oo ) -> ( (π ` x ) / ( x / ( log ` x ) ) ) e. RR )
83	82	recnd 9668	. . . . . 6 |- ( x e. ( 1 (,) +oo ) -> ( (π ` x ) / ( x / ( log ` x ) ) ) e. CC )
84	83	adantl 467	. . . . 5 |- ( ( T. /\ x e. ( 1 (,) +oo ) ) -> ( (π ` x ) / ( x / ( log ` x ) ) ) e. CC )
85		eqid 2429	. . . . 5 |- ( x e. ( 1 (,) +oo ) |-> ( (π ` x ) / ( x / ( log ` x ) ) ) ) = ( x e. ( 1 (,) +oo ) |-> ( (π ` x ) / ( x / ( log ` x ) ) ) )
86	84, 85	fmptd 6061	. . . 4 |- ( T. -> ( x e. ( 1 (,) +oo ) |-> ( (π ` x ) / ( x / ( log ` x ) ) ) ) : ( 1 (,) +oo ) --> CC )
87	12	a1i 11	. . . 4 |- ( T. -> ( 1 (,) +oo ) C_ RR )
88	15	a1i 11	. . . 4 |- ( T. -> 2 e. RR )
89	86, 87, 88	rlimresb 13607	. . 3 |- ( T. -> ( ( x e. ( 1 (,) +oo ) |-> ( (π ` x ) / ( x / ( log ` x ) ) ) ) ~~> r 1 <-> ( ( x e. ( 1 (,) +oo ) |-> ( (π ` x ) / ( x / ( log ` x ) ) ) ) |` ( 2 [,) +oo ) ) ~~> r 1 ) )
90	77, 89	mpbird 235	. 2 |- ( T. -> ( x e. ( 1 (,) +oo ) |-> ( (π ` x ) / ( x / ( log ` x ) ) ) ) ~~> r 1 )
91	90	trud 1446	1 |- ( x e. ( 1 (,) +oo ) |-> ( (π ` x ) / ( x / ( log ` x ) ) ) ) ~~> r 1

Syntax hints:   <-> wb 187   /\ w3a 982   = wceq 1437   T. wtru 1438   e. wcel 1870   =/= wne 2625   C_ wss 3442   class class class wbr 4426   |-> cmpt 4484   |` cres 4856   ` cfv 5601  (class class class)co 6305   CCcc 9536   RRcr 9537   0cc0 9538   1c1 9539   x. cmul 9543   +oocpnf 9671   RR*cxr 9673   < clt 9674   <_ cle 9675   / cdiv 10268   NNcn 10609   2c2 10659   NN0cn0 10869   RR+crp 11302   (,)cioo 11635   [,)cico 11637   ~~> r crli 13527   logclog 23369   thetaccht 23880  πcppi 23883

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-inf2 8146  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615  ax-pre-sup 9616  ax-addf 9617  ax-mulf 9618

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-iin 4305  df-disj 4398  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-se 4814  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-isom 5610  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-of 6545  df-om 6707  df-1st 6807  df-2nd 6808  df-supp 6926  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-2o 7191  df-oadd 7194  df-er 7371  df-map 7482  df-pm 7483  df-ixp 7531  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-fsupp 7890  df-fi 7931  df-sup 7962  df-inf 7963  df-oi 8025  df-card 8372  df-cda 8596  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-div 10269  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-7 10673  df-8 10674  df-9 10675  df-10 10676  df-n0 10870  df-z 10938  df-dec 11052  df-uz 11160  df-q 11265  df-rp 11303  df-xneg 11409  df-xadd 11410  df-xmul 11411  df-ioo 11639  df-ioc 11640  df-ico 11641  df-icc 11642  df-fz 11783  df-fzo 11914  df-fl 12025  df-mod 12094  df-seq 12211  df-exp 12270  df-fac 12457  df-bc 12485  df-hash 12513  df-shft 13109  df-cj 13141  df-re 13142  df-im 13143  df-sqrt 13277  df-abs 13278  df-limsup 13504  df-clim 13530  df-rlim 13531  df-o1 13532  df-lo1 13533  df-sum 13731  df-ef 14099  df-e 14100  df-sin 14101  df-cos 14102  df-pi 14104  df-dvds 14284  df-gcd 14443  df-prm 14594  df-pc 14750  df-struct 15086  df-ndx 15087  df-slot 15088  df-base 15089  df-sets 15090  df-ress 15091  df-plusg 15165  df-mulr 15166  df-starv 15167  df-sca 15168  df-vsca 15169  df-ip 15170  df-tset 15171  df-ple 15172  df-ds 15174  df-unif 15175  df-hom 15176  df-cco 15177  df-rest 15280  df-topn 15281  df-0g 15299  df-gsum 15300  df-topgen 15301  df-pt 15302  df-prds 15305  df-xrs 15359  df-qtop 15364  df-imas 15365  df-xps 15367  df-mre 15443  df-mrc 15444  df-acs 15446  df-mgm 16439  df-sgrp 16478  df-mnd 16488  df-submnd 16534  df-mulg 16627  df-cntz 16922  df-cmn 17367  df-psmet 18897  df-xmet 18898  df-met 18899  df-bl 18900  df-mopn 18901  df-fbas 18902  df-fg 18903  df-cnfld 18906  df-top 19852  df-bases 19853  df-topon 19854  df-topsp 19855  df-cld 19965  df-ntr 19966  df-cls 19967  df-nei 20045  df-lp 20083  df-perf 20084  df-cn 20174  df-cnp 20175  df-haus 20262  df-cmp 20333  df-tx 20508  df-hmeo 20701  df-fil 20792  df-fm 20884  df-flim 20885  df-flf 20886  df-xms 21266  df-ms 21267  df-tms 21268  df-cncf 21806  df-limc 22698  df-dv 22699  df-log 23371  df-cxp 23372  df-em 23783  df-cht 23886  df-vma 23887  df-chp 23888  df-ppi 23889  df-mu 23890

This theorem is referenced by: (None)